(Belatedly) The Wikipedia article Polylogarithm (https://en.wikipedia.org/wiki/Polylogarith) shows several relationships among the polylogarithms, the Bernoulli numbers, Riemann and Hurwitz zeta functions, and the Fermi-Dirac and Bose-Einstein distributions. In https://oeis.org/A131758, I pointed out relations among several special number sequences and associated polynomials, their interpolations, the deformed Todd operator, and the FD and BE distributions. Taking the Mellin transform of the distributions gives the polylogarithm over two different domains. Hyper (cluster) polylogarithms have also popped up in recent papers on scattering amplitudes.

]]>Cut and paste error: In my derivative formula, obviously one should be .

]]>Looking at the Bernoulli polynomials rather than just the #s seems to collate formulas. Ramanujan’s Master Formula gives the Hurwitz zeta function as essentially an entire function interpolation of the Bernoulli polynomials with e.g.f. , for which the moments are the Bernoulli numbers with e.g.f. with . This e.g.f. reflects the Bernoulli polynomials’ umbral derivational property , which extends formally term by term to any Taylor series as . Defining destroys this derivational property and redefines all higher degree Bernoulli polynomials. Rather, , and this is the consistent with the RMT/Mellin interpolation of the Bernoulli polynomials to the Hurwitz zeta function , giving . In the integrand of the MT interpolator, is the correct expression, which differs only in the term linear in from , i.e., with all odd index values vanishing but for . Confer https://mathoverflow.net/questions/353282/an-analytic-continuation-of-power-series-coefficients/379477#379477. Reprising, the Riemann zeta function can be regarded as the Mellin interpolation of the sequence . Summation formulas with the ‘Todd’ operator retain a nice form as well (cf. https://mathoverflow.net/questions/380142/intuitive-explanation-why-shadow-operator-frac-ded-1-connects-logarithms/380189#380189).

]]>Hardy rated himself a C mathematician compared to Ramanujan as an A. (That puts me off the alphabet. I don’t use the term genius– in some sense it marginalizes the passion, the diligence and dedication, even obsession, of the masters.)

]]>Great Srinivasa Ramanujan! is my source of inspiration and was a genius, as Littlewood said, comparable to a Jacobi or an Euler

]]>\[ \int^n_m f(z)dz=\sum_{k=m}^nf(k)-\frac{f(n)-f(m)}{2}-\sum_{h=1}^{\lfloor p/2\rfloor}\frac{B_{2h}}{(2h)!}(f^{(2h-1)}(n)-f^{(2h-1)}(m))-R_p \]

where \(R_p=\mathcal{O}(\int_m^n |f^{(p)}(z)|dz) \).

let \( \alpha=\mathcal{Re}[s-B]$ and $g(z)=\zeta(z)\frac{N^{z-s}F(z-s)}{z-s}: \)

\[ \int^{\alpha+i \infty}_{\alpha-i \infty} g(z)dz=\sum_{k=-\infty}^{+ \infty}g(\alpha+i k)+\mathcal{O}(\int_{\alpha-i \infty}^{\alpha+i \infty} |g^{(p)}(z)|dz) \]

at this point I have doubts, I could change the integration line by choosing a \( \alpha \) that \( \forall k \) neglects \( g (\alpha + ik) \) but anyway I don’t know how to show that \( \mathcal {O} (\int_ {\alpha-i \infty} ^ {\alpha + i \infty} | g ^ {(p)} (z) | dz) = \mathcal {O} (N ^ {- B}) \)

Introduction to Divergent Series of Integers

https://divergent.thinkific.com/courses/dsi-101 ]]>